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Symmetric and Asymmetric Distributions • Emilio Gómez-Déniz Symmetric and Asymmetric Distributions Symmetric and Asymmetric Distributions and Symmetric • Emilio Gómez-Déniz • Emilio Symmetric and Asymmetric Distributions Edited by Emilio Gómez-Déniz Printed Edition of the Special Issue Published in Symmetry www.mdpi.com/journal/symmetry Symmetric and Asymmetric Distributions Symmetric and Asymmetric Distributions Theoretical Developments and Applications Editor Emilio Gomez´ -Deni´ z MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editor Emilio Gomez-D´´ eniz Department of Quantitative Methods and TIDES Institute, University of Las Palmas de Gran Canaria Spain Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Symmetry (ISSN 2073-8994) (available at: https://www.mdpi.com/journal/symmetry/special issues/Symmetric Asymmetric Distributions Theoretical Developments Applications). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03936-646-0 (Hbk) ISBN 978-3-03936-647-7 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editor .............................................. vii Preface to ”Symmetric and Asymmetric Distributions” ........................................ ix Emilio G´omez-D´eniz, Yuri A. Iriarte, Enrique Calder´ın-Ojeda and H´ector W. G´omez Modified Power-Symmetric Distribution Reprinted from: Symmetry 2019, 11, 1410, doi:10.3390/sym11111410 ................. 1 F´abio V. J. Silveira, Frank Gomes-Silva, C´ıcero C. R. Brito, Moacyr Cunha-Filho, Felipe R. S. Gusm˜ao and S´ılvio F. A. Xavier-Junior ´ Normal-G Class of Probability Distributions: Properties and Applications Reprinted from: Symmetry 2019, 11, 1407, doi:10.3390/sym11111407 ................. 17 H´ector J. G´omez, Diego I. Gallardo and Osvaldo Venegas Generalized Truncation Positive Normal Distribution Reprinted from: Symmetry 2019, 11, 1361, doi:10.3390/sym11111361 ................. 35 Tiago M. Magalh˜aes, Diego I. Gallardo and H´ector W. G´omez Skewness of Maximum Likelihood Estimators in the Weibull Censored Data Reprinted from: Symmetry 2019, 11, 1351, doi:10.3390/sym11111351 ................. 53 Guillermo Mart´ınez-Fl´orez, Inmaculada Barranco-Chamorro, Heleno Bolfarine and H´ector W. Gomez ´ Flexible Birnbaum–Saunders Distribution Reprinted from: Symmetry 2019, 11, 1305, doi:10.3390/sym11101305 ................. 63 Neveka M. Olmos, Osvaldo Venegas, Yolanda M. G´omez and Yuri Iriarte An Asymmetric Distribution with Heavy Tails and Its Expectation–Maximization (EM) Algorithm Implementation Reprinted from: Symmetry 2019, 11, 1150, doi:10.3390/sym11091150 ................. 81 Yolanda M. G´omez, Emilio G´omez-D´eniz, Osvaldo Venegas, Diego I. Gallardo and H´ector W. Gomez ´ An Asymmetric Bimodal Distribution with Application to Quantile Regression Reprinted from: Symmetry 2019, 11, 899, doi:10.3390/sym11070899 ................. 95 Barry C. Arnold, H´ector W. G´omez, H´ector Varela and Ignacio Vidal Univariate and Bivariate Models Related to the GeneralizedEpsilon–Skew–Cauchy Distribution Reprinted from: Symmetry 2019, 11, 794, doi:10.3390/sym11060794 .................107 V. J. Garc´ıa, M. Martel–Escobar and F. J. V´azquez–Polo A Note on Ordering Probability Distributions by Skewness Reprinted from: Symmetry 2018, 10, 286, doi:10.3390/sym10070286 .................121 v About the Editor Emilio Gomez´ -D´eniz (M.S. in Mathematics, M.S. in Economics, and Ph.D. in Economics) is professor of Mathematics at the University of Las Palmas de Gran Canaria (Spain), where he has taught and conducted research for more than 30 years. He is also member of the Institute of Tourism and Sustainable Economic Development (TIDES). His main lines of research focus on distribution theory, Bayesian statistics, robustness, and Bayesian applications in economics, with emphasis in actuarial statistics, tourism, education, sports, electronic engineering, health economics, etc. He has published more than 150 articles, most of them with JCR impact factor according to Web of Science and his work has been cited more than 1000 times on Google Scholar with an h-index of 16 and an i10 index of 35. He is currently the editor or member of the editorial board of the Spanish Journal of Statistics (SJS), Chilean Journal of Statistics, Mathematical Problems in Engineering, Risks, Risk Magazine, and Journal of Financial and Risk Management (Reviewer Board) and Statistical Methodology and Mathematical Reviews in the past. He has also acted and regularly acts as reviewer for more than 100 international journals. His academic leadership is further reflected through collaborative research with more than 50 leading international scientists worldwide. He is the author of many books on mathematics education and research, including ”Actuarial Statistics. Theory and Applications”, Prentice Hall (in Spanish). He has organized and participated in many international conferences and seminars. In addition, he has visited, as guest researcher and speaker, the University of Melbourne, the University of Kuwait, the University of Cantabria, the University of Antofagasta (Chile), the University of Barcelona, and the University of Granada, among others. In 2008, he was awarded the prestigious IV Julio Castelo Matran´ International Insurance Award, sponsored by the Mapfre Foundation and also recognized by his university for excellence in teaching. vii Preface to ”Symmetric and Asymmetric Distributions” This Special Issue contains nine chapters selected after a comprehensive peer review process. Each chapter exclusively adheres to the topic specified in this Special Issue. The Associate Editor wants to especially thank all the authors who made this Special Issue possible, and all the anonymous reviewers for their altruistic effort and help in reviewing and for their excellent suggestions and critical reviews of the submitted manuscripts. All the chapters are written in the format of a research article for scholarly journals, i.e., title, abstract, and development, with an extensive bibliography at the end of the paper. All of them include real applications that will undoubtedly be useful for other researchers and graduate students who conduct similar research. I express my gratitude to MPDI for publishing this book, to Ms. Celina Si, Section Managing Editor, for her work and patience, and last but not least to my family for their support and cooperation. Emilio Gomez´ -D´eniz Editor ix S symmetry S Article Modified Power-Symmetric Distribution Emilio Gómez-Déniz 1, Yuri A. Iriarte 2, Enrique Calderín-Ojeda 3 and Héctor W. Gómez 2,* 1 Department of Quantitative Methods in Economics and TIDES Institute, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain; [email protected] 2 Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile; [email protected] 3 Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Melbourne, VIC 3010, Australia; [email protected] * Correspondence: [email protected] Received: 2 October 2019; Accepted: 13 November 2019; Published: 15 November 2019 Abstract: In this paper, a general class of modified power-symmetric distributions is introduced. By choosing as symmetric model the normal distribution, the modified power-normal distribution is obtained. For the latter model, some of its more relevant statistical properties are examined. Parameters estimation is carried out by using the method of moments and maximum likelihood estimation. A simulation analysis is accomplished to study the performance of the maximum likelihood estimators. Finally, we compare the efficiency of the modified power-normal distribution with other existing distributions in the literature by using a real dataset. Keywords: maximum likelihood; kurtosis; power-normal distribution 1. Introduction Over the last few years, the search for flexible probabilistic families capable of modeling different levels of bias and kurtosis has been an issue of great interest in the field of distributions theory. This was mainly motivated by the seminal work of Azzalini [1]. In that paper, the probability density function (pdf) of a skew-symmetric distribution was introduced. The expression of this density is given by g(z; λ)=2 f (z)G(λz), z, λ ∈ R, (1) where f (·) is a symmetric pdf about zero; G(·) is an absolutely continuous distribution function, which is also symmetric about zero; and λ is a parameter of asymmetry. For the case where f (·) is the standard normal density (from now on, we reserve the symbol φ for this function), and G(·) is the standard normal cumulative distribution function (henceforth, denoted by Φ), the so-called skew-normal (SN) distribution with density φ ( λ)= φ( )Φ(λ ) λ ∈ R Z z; 2 z z , z, , (2) is obtained. We use the notation Z ∼SN(λ) to denote the random variable Z with pdf given by Equation (2). A generalization
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